What do non-relativistic CFTs tell us about Lifshitz spacetimes?

We study the reconstructability of (d+2)-dimensional bulk spacetime from (d+1)-dimensional boundary data, particularly concentrating on backgrounds which break (d+1)-dimensional Lorentz invariance. For a large class of such spacetimes, there exist null geodesics which do not reach the boundary. Therefore classically we expect some information is trapped in the bulk and thus invisible at the boundary. We show that this classical intuition correctly predicts the quantum situation: whenever there are null geodesics which do not reach the boundary, there are also "trapped scalar modes" whose boundary imprint is exponentially suppressed. We use these modes to show that no smearing function exists for pure Lifshitz spacetime, nor for any flow which includes a Lifshitz region. Indeed, for any (planar) spacetime which breaks (d+1)-dimensional Lorentz invariance at any radius, we show that local boundary data cannot reconstruct complete local bulk data.

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